Gruber P. Convex and Discrete Geometry

15 Combinatorial Theory of Convex Polytopes 271
For additional information the reader may consult the books of Grünbaum [453],
Ziegler [1045], Richter-Gebert [833] and Matouˇsek [695] and the surveys of Bayer
and Lee [83] and Kalai [561].
Steinitz’ Representation Theorem for Convex Polytopes in E 3
We shall prove the following basic result.
Theorem 15.6. Let G be a planar 3-connected graph. Then G is isomorphic to the
edge graph of a convex polytope in E 3 .
Proof. For terminology, see Sect. 34.1. An application of the Koebe–Brightwell–
Scheinerman theorem 34.1 together with a suitable Möbius transformation and stereographic
projection shows that there is a primal-dual circle representation of G on the
unit sphere S 2 with the following properties.
The country circle of the outer country is the equator.
All other country circles are in the southern hemisphere.
Three of these touch the equator at points which are 2π 3 apart.
From now on we use this primal-dual circle representation of G.
For each country circle choose a closed halfspace with the country circle in its
boundary plane such that the halfspace of the equator circle contains the south pole
of S 2 and the other halfspaces the origin o. The intersection of these halfspaces is
then a convex polytope P. The countries, resp. the country circles of G correspond
to the facets of P. Letv ∈ S 2 be a vertex (of the representation) of G on S 2 .The
country circles of the countries with vertex v form a ring around v like a string
of beads, possibly of different sizes, on the vertex circle of v. The latter intersects
these country circles orthogonally. All other country circles are outside this ring.
This shows that the boundary planes of the halfspaces corresponding to the circles
of the ring meet in a point v P, say, radially above v and v P is an interior point of all
other halfspaces. Thus v P is a vertex of P. IfC is a country circle of the ring, the
facet F of P determined by C has two edges which contain v P and are tangent to C
and thus to S 2 .Ifvw···z is a cycle of G around C, then v PwP ···z P is a cycle of
edges of F and thus is the cycle of edges of F. Each edge of F is tangent to S 2 at
the point where it touches the incircle C of F. Since, by construction, the edge graph
of P is obtained by radial projection of the representation of G (in S 2 ), the proof is
complete. ⊓⊔
Different Ways to Represent a Convex Polytope
The above proof of the Steinitz representation theorem shows that any convex polytope
in E 3 is (combinatorially) isomorphic to a convex polytope all edges of which
touch the unit ball B 3 . A far-reaching generalization of this result is due to Schramm
[914] in which B 3 is replaced by any smooth and strictly convex body in E 3 .